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arXiv:1002.2471v2 [hep-ph] 28 Apr 2010

Analysis of the

3

2

+heavy and doubly heavy baryon states with QCD

sum rules

Zhi-Gang Wang1

Department of Physics, North China Electric Power University, Baoding 071003, P. R.

China

Abstract

In this article, we study the

bb, Ω∗

bb, Σ∗

c, Ξ∗

c, Ω∗

responding3

2

make reasonable predictions for their masses.

3

2

+heavy and doubly heavy baryon states Ξ∗

band Ω∗

bby subtracting the contributions from the cor-

−heavy and doubly heavy baryon states with the QCD sum rules, and

cc, Ω∗

cc,

Ξ∗

c, Σ∗

b, Ξ∗

PACS number: 14.20.Lq, 14.20.Mr

Key words: Heavy baryon states, QCD sum rules

1 Introduction

In 2006, the Babar collaboration reported the first observation of the3

state Ω∗

c→ Ωcγ [1]. By now, the

Ξ+

c), and the

22

established [2].

In 2008, the D0 collaboration reported the first observation of the doubly strange

baryon state Ω−

[3]. The experimental value MΩ−

than the most theoretical calculations [4, 5, 6, 7, 8, 9, 10, 11, 12, 14]. However, the

CDF collaboration did not confirm the measured mass [16], i.e. they observed the mass

of the Ω−

bis about (6.0544 ± 0.0068 ± 0.0009)GeV, which is consistent with the most

theoretical calculations. On the other hand, the theoretical prediction MΩ0

[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] is consistent with the experimental data MΩ0

(2.6975 ± 0.0026)GeV [2]. The S-wave bottom baryon states are far from complete, only

the Λb, Σb, Σ∗

In 2002, the SELEX collaboration reported the first observation of a signal for the

doubly charm baryon state Ξ+

confirmed later by the same collaboration in the decay mode Ξ+

mass MΞ= (3518.9±0.9)MeV [18]. However, the Babar and Belle collaborations have not

observed any evidence for the doubly charm baryon states in e+e−annihilations [19, 20].

No experimental evidences for the3

2

have been several approaches to deal with the doubly heavy baryon masses, such as the

relativistic quark model [21, 22], the non-relativistic quark model [14, 23, 24, 25], the

three-body Faddeev method [5], the potential approach combined with the QCD sum

rules [26], the quark model with AdS/QCD inspired potential [27], the MIT bag model

[28], the full QCD sum rules [29, 30], the Feynman-Hellmann theorem and semiempirical

mass formulas [31], and the effective field theories [32], etc.

2

+heavy baryon

cin the radiative decay Ω∗

1

+and

1

2

+antitriplet states (Λ+

c,Σ∗

c,

c,Ξ0

3

+sextet states (Ωc,Σc,Ξ′c) and (Ω∗

c,Ξ∗

c) have been well

bin the decay channel Ω−

b→ J/ψ Ω−in p¯ p collisions at√s = 1.96 TeV

b= (6.165 ± 0.010 ± 0.013) GeV is about 0.1GeV larger

c≈ 2.7GeV

c=

b, Ξb, Ωbhave been observed [2].

ccin the charged decay mode Ξ+

cc→ Λ+

cc→ pD+K−with measured

cK−π+[17], and

+doubly heavy baryon states are observed [2]. There

1E-mail:wangzgyiti@yahoo.com.cn.

1

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The charm and bottom baryon states which contain one (two) heavy quark(s) are

particularly interesting for studying dynamics of the light quarks in the presence of the

heavy quark(s), and serve as an excellent ground for testing predictions of the quark

models and heavy quark symmetry. On the other hand, the QCD sum rules is a powerful

theoretical tool in studying the ground state heavy baryon states [33, 34, 35].

In the QCD sum rules, the operator product expansion is used to expand the time-

ordered currents into a series of quark and gluon condensates which parameterize the

long distance properties of the QCD vacuum. Based on the quark-hadron duality, we can

obtain copious information about the hadronic parameters at the phenomenological side

[33, 34, 35]. There have been several works on the masses of the heavy baryon states with

the full QCD sum rules and the QCD sum rules in the heavy quark effective theory (one

can consult Ref.[36] for more literatures).

In Ref.[37], Jido et al introduce a novel approach based on the QCD sum rules to

separate the contributions of the negative-parity light flavor baryons from the positive-

parity light flavor baryons, as the interpolating currents may have non-vanishing couplings

to both the negative- and positive-parity baryons [38]. Before the work of Jido et al, Bagan

et al take the infinite mass limit for the heavy quarks to separate the contributions of the

positive and negative parity heavy baryon states unambiguously [39].

In Refs.[40, 41, 42], we study the heavy baryon states ΩQ, Ξ′

with the full QCD sum rules, and observe that the pole residues of the3

from the sum rules with different tensor structures are consistent with each other, while the

pole residues of the1

2

differ from each other greatly. In Refs.[36, 43], we follow Ref.[37] and study the masses and

pole residues of the1

2

contributions of the negative parity heavy baryon states to overcome the embarrassment.

Those pole residues are important parameters in studying the radiative decays Ω∗

Ξ∗

Q→ ΣQγ [42, 44], etc. In Ref.[45], we extend our previous works to

study the1

2

In this article, we study the3

2

Ω∗

bby subtracting the contributions from the corresponding

3

2

The article is arranged as follows: we derive the QCD sum rules for the masses and

the pole residues of the heavy and doubly heavy baryon states Ξ∗

Ω∗

bin Sect.2; in Sect.3, we present the numerical results and discussions;

and Sect.4 is reserved for our conclusions.

Q, ΣQ, Ω∗

Q, Ξ∗

+heavy baryons

Qand Σ∗

Q

2

+heavy baryons from the sum rules with different tensor structures

+heavy baryon states ΩQ, Ξ′

Q, ΣQ, ΛQand ΞQby subtracting the

Q→ ΩQγ,

Q→ Ξ′

Qγ and Σ∗

+doubly heavy baryon states ΞQQand ΩQQwith the full QCD sum rules.

+heavy and doubly heavy baryon states Ξ∗

c, Ξ∗

−heavy and doubly heavy baryon states with the QCD sum rules.

cc, Ω∗

cc, Ξ∗

bb,

bb, Σ∗

c, Ω∗

c, Σ∗

b, Ξ∗

band Ω∗

cc, Ω∗

cc, Ξ∗

bb, Ω∗

bb, Σ∗

c, Ξ∗

c,

c, Σ∗

b, Ξ∗

band Ω∗

2 QCD sum rules for the baryon states Ω∗

and Σ∗

Q

QQ, Ξ∗

QQ, Ω∗

Q, Ξ∗

Q

The3

2

+heavy and doubly heavy baryon states Ω∗

QQ, Ξ∗

QQ, Ω∗

Ω∗

Q

µ (x), J

Q, Ξ∗

Ξ∗

µ (x) and J

Qand Σ∗

Qcan be inter-

Σ∗

Q

µ (x) respec-polated by the following currents J

Ω∗

µ

QQ

(x), J

Ξ∗

µ

QQ

(x), J

Q

2

Page 3

tively,

J

Ω∗

µ

Ξ∗

µ

QQ

(x)=ǫijkQT

i(x)CγµQj(x)sk(x),

ǫijkQT

i(x)CγµQj(x)qk(x),

ǫijksT

i(x)Cγµsj(x)Qk(x),

ǫijkqT

i(x)Cγµsj(x)Qk(x),

ǫijkuT

i(x)Cγµdj(x)Qk(x),

J

QQ

(x)=

J

Ω∗

µ (x)

Ξ∗

Q

µ (x)

Σ∗

Q

µ (x)

Q

=

J

=

J

= (1)

where the Q represents the heavy quarks c and b, the i, j and k are color indexes, and

the C is the charge conjunction matrix. In the heavy quark limit, the heavy and doubly

heavy baryon states can be described by the diquark-quark model [26].

The corresponding3

2

the currents J−

[37], where the J+

µdenotes the currents J

µ

The correlation functions Π±

µν(p) are defined by

?

The currents J±

2

[38], i.e.

−heavy and doubly heavy baryon states can be interpolated by

µ= iγ5J+

Ω∗

QQ

(x), J

µ

µbecause multiplying iγ5to the J+

µchanges the parity of the J+

Ω∗

Q

µ (x), J

µ (x) and J

µ

Ξ∗

QQ

(x), J

Ξ∗

Q

Σ∗

µ (x).

Q

Π±

µν(p)=id4xeip·x?0|T?J±

±baryon states B∗

µ(x)¯J±

ν(0)?|0?.

±and the1

(2)

µ(x) couple to both the3

2

±baryon states B±

?0|J+

?0|J+

µ(0)|B∗

µ(0)|B±(p)??B±(p)|¯J+

±(p)??B∗

±(p)|¯J+

ν(0)|0?

ν(0)|0?

=−γ5?0|J−

−γ5?0|J−

µ(0)|B∗

µ(0)|B±(p)??B±(p)|¯J−

±(p)??B∗

±(p)|¯J−

ν(0)|0?γ5,

ν(0)|0?γ5,= (3)

where

?0|J±

?0|J±

µ(0)|B∗

µ(0)|B∓(p)?

±(p)?=λ±Uµ(p,s),

?

=λ∗

γµ− 4pµ

M∗

?

U(p,s), (4)

the λ±and λ∗are the pole residues and M∗are the masses, and the spinor U(p,s) satisfies

the usual Dirac equation (?p − M∗)U(p) = 0.

The Π±

µν(p) have the following relation

Π−

µν(p)=−γ5Π+

µν(p)γ5.(5)

We insert a complete set of intermediate baryon states with the same quantum numbers

as the current operators J±

representation [33, 34]. After isolating the pole terms of the lowest states of the heavy

and doubly heavy baryons, we obtain the following result [37]:

µ(x) into the correlation functions Π+

µν(p) to obtain the hadronic

Π+

µν(p)=−λ2

−Π+(p)gµν+ ··· ,

+

?p + M+

M2

+− p2gµν− λ2

−

?p − M−

M2

−− p2gµν+ ··· ,

=(6)

3

Page 4

where the M± are the masses of the lowest states with parity ± respectively, and the

λ±are the corresponding pole residues (or couplings). In calculations, we have used the

following equations,

?

?

In this article, we choose the tensor structure gµνfor analysis, the1

no contaminations.

If we take ? p = 0, then

?

s

Uµ(p,s)Uν(p,s)=(?p + MB∗)−gµν+γµγν

3

+2pµpν

3M2

B∗

−pµγν− pνγµ

3MB∗

?

,

s

U(p,s)U(p,s)= ?p + M∗. (7)

2

±baryon states have

limitǫ→0ImΠ+(p0+ iǫ)

π

=λ2

+

γ0+ 1

2

δ(p0− M+) + λ2

−

γ0− 1

2

δ(p0− M−) + ···

=γ0A(p0) + B(p0) + ··· ,(8)

where

A(p0)=

1

2

1

2

?λ2

?λ2

+δ(p0− M+) + λ2

−δ(p0− M−)?,

B(p0)=

+δ(p0− M+) − λ2

−δ(p0− M−)?, (9)

the A(p0) + B(p0) and A(p0) − B(p0) contain the contributions from the positive-parity

states and negative-parity baryon states respectively.

We calculate the light quark parts of the correlation functions Π+

space and use the momentum space expression for the heavy quark propagators, i.e. we

take

µν(p) in the coordinate

Sij(x)=

iδij?x

2π2x4−δijms

i

32π2x2Gij

?

+π2

4π2x2−δij

12?¯ ss? +iδij

48ms?¯ ss??x

−

µν(x)[?xσµν+ σµν?x] + ··· ,

?

?k − mQ

3?αsGG

π

(k2− m2

Sij

Q(x)=

i

(2π)4

d4ke−ik·x

δij

−gsGαβ

ij

4

σαβ(?k + mQ) + (?k + mQ)σαβ

(k2− m2

?

Q)2

?δijmQ

k2+ mQ?k

Q)4+ ···, (10)

where ?αsGG

quark parts into the momentum space in D dimensions, take ? p = 0, and use the dispersion

relation to obtain the spectral densities ρA(p0) and ρB(p0) (which correspond to the tensor

structures γ0and 1 respectively) at the level of quark-gluon degrees of freedom. Finally

we introduce the weight functions exp−p2

rules,

?

π

? = ?αsGαβGαβ

π

?, then resort to the Fourier integral to transform the light

?

0

T2

?

, p2

0exp

?

−p2

0

T2

?

, and obtain the following sum

λ2

+exp−M2

T2

+

?

=

?√s0

∆

dp0

?ρA(p0) + ρB(p0)?exp

4

?

−p2

T2

0

?

,(11)

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λ2

+M2

+exp

?

−M2

T2

+

?

=

?√s0

∆

dp0

?ρA(p0) + ρB(p0)?p2

0exp

?

−p2

T2

0

?

, (12)

where the s0are the threshold parameters, T2are the Borel parameters, and ∆ = 2mQ+

ms, 2mQ, mQ+ 2ms, mQ+ ms and mQ in the channels Ω∗

respectively. The spectral densities ρA(p0) and ρB(p0) at the level of quark-gluon degrees

of freedom are given explicitly in the Appendix.

QQ, Ξ∗

QQ, Ω∗

Q, Ξ∗

Qand Σ∗

Q

3 Numerical results and discussions

The input parameters are taken to be the standard values ?¯ qq? = −(0.24 ± 0.01GeV)3,

?¯ ss? = (0.8 ± 0.2)?¯ qq?, ?¯ qgsσGq? = m2

[46, 47], ?αsGG

0.10)GeV and mb= (4.7 ± 0.1)GeV [2] at the energy scale µ = 1GeV.

The value of the gluon condensate ?αsGG

changes greatly [35]. At the present case, the gluon condensate makes tiny contribution,

the updated value ?αsGG

(0.012±0.004)GeV4[47] lead to a tiny difference and can be neglected safely. The values of

the quark condensates determined from the Gell-Mann-Oakes-Renner relation, the spectral

functions of the τ decay, and the QCD sum rules for baryon masses are consistent with

each other within uncertainties [46], we usually take the value from the Gell-Mann-Oakes-

Renner relation in the QCD sum rules [47]. For the mixed condensates, we take the

value from the QCD sum rules for the baryonic resonances, which is still accepted in

the literatures [46, 47]. Those values are not accurate, and there are much room for

improvement; the update of the vacuum condensates should be combined with a more

delicate procedure in dealing with the perturbative and non-perturbative contributions,

and beyond the present work.

The Q-quark masses appearing in the perturbative terms are usually taken to be

the pole masses in the QCD sum rules, while the choice of the mQin the leading-order

coefficients of the higher-dimensional terms is arbitrary [35, 48]. The MS mass mc(m2

relates with the pole mass ˆ m through the relation mc(m2

this article, we take the approximation mc≈ ˆ m without the αscorrections for consistency.

The value listed in the Particle Data Group is mc(m2

to take mc= mc(1GeV2) = (1.35 ± 0.10)GeV. The value of the mbcan be understood

analogously.

In calculation, we also neglect the contributions from the perturbative O(αn

tions. Those perturbative corrections can be taken into account in the leading logarithmic

approximations through the anomalous dimension factors. After the Borel transform, the

effects of those corrections are to multiply each term on the operator product expansion

?αs(T2)

terpolating current J(x), and the ΓOnis the anomalous dimension of the local operator

On(0), which governs the evolution of the vacuum condensate ?On(0)?µwith the energy

scale through the re-normalization group equation.

If the perturbative O(αs) corrections and the anomalous dimension factors are taken

0?¯ qq?, ?¯ sgsσGs? = m2

0?¯ ss?, m2

0= (0.8 ± 0.2)GeV2

π

? = (0.012 ± 0.004)GeV4[47], ms = (0.14 ± 0.01)GeV, mc = (1.35 ±

π

? has been updated from time to time, and

π

? = (0.023 ± 0.003)GeV4[35] and the standard value ?αsGG

π

? =

c)

c) = ˆ m

?

1 +CFαs(m2

c)

π

+ ···

?−1. In

c) = 1.27+0.07

−0.11GeV [2], it is reasonable

s) correc-

side by the factor,

αs(µ2)

?2ΓJ−ΓOn, where the ΓJ is the anomalous dimension of the in-

5